Wizmic microtemperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 microtemperaments which temper out the wizma (monzo: [-6 -8 2 5⟩, ratio: 420175/419904).

Temperaments discussed elsewhere are:

• Superpyth (+64/63 or 245/243) → Archytas clan
• Gariwizmic (+33554432/33480783) → Garischismic clan
• Cloudtone (+81/80) → Meantone family
• Tsaharuk (+32805/32768) → Schismatic family
• Hemiseven (+1029/1024) → Gamelismic clan
• Septiquarter (+5120/5103) → Hemifamity temperaments
• Hendecatonic (+6144/6125 or 10976/10935) → Porwell temperaments
• Wizard (+225/224) → Marvel temperaments
• Oolong (+126/125) → Starling temperaments
• Ennealimmal (+2401/2400 or 4375/4374) → Septiennealimmal clan
• Fifthplus (+65625/65536) → Horwell temperaments
• Octowerck (+321489/320000) → Varunismic temperaments
• Sengagen (+3136/3125) → Hemimean clan
• Sqrtphi (+15625/15552 or 16875/16807) → Kleismic family
• Quinwell (+2109375/2097152) → Semicomma family
• Decavish (+2202927104/2197265625) → Vishnuzmic family

Considered below are qak, tokko, tertiathirds, and witcher.

Qak

Qak tempers out the meter in addition to the wizma, and may be described as the 50 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 703125/702464

Mapping: [⟨1 -14 -3 -20], ⟨0 41 14 60]]

mapping generators: ~2, ~125/96

Optimal tunings:

• WE: ~2 = 1200.0566 ¢, ~125/96 = 456.1655 ¢

error map: ⟨+0.057 +0.039 -0.166 -0.027]

• CWE: ~2 = 1200.0000 ¢, ~125/96 = 456.1450 ¢

error map: ⟨0.000 -0.009 -0.283 -0.125]

Optimal ET sequence: 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd

Badness (Sintel): 0.741

Tokko

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Tokko (5-limit).

Tokko tempers out the mitonisma in addition to the wizma, and may be described as the 5 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 5250987/5242880

Mapping: [⟨1 -1 -11 4], ⟨0 13 67 -6]]

mapping generators: ~2, ~147/128

Optimal tunings:

• WE: ~2 = 1200.0850 ¢, ~147/128 = 238.6157 ¢

error map: ⟨+0.085 -0.036 +0.004 -0.180]

• CWE: ~2 = 1200.0000 ¢, ~147/128 = 238.5997 ¢

error map: ⟨0.000 -0.159 -0.133 -0.424]

Optimal ET sequence: 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd

Badness (Sintel): 1.12

Tertiathirds

Tertiathirds tempers out the quasiorwellisma, 29360128/29296875 in the 7-limit, and may be described as the 121 & 149 temperament. It splits the interval of 5/4 into three 14/13 generators.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 29360128/29296875

Mapping: [⟨1 -4 2 -6], ⟨0 52 3 82]]

mapping generators: ~2, ~3375/3136

Optimal tunings:

• WE: ~2 = 1199.9305 ¢, ~3375/3136 = 128.8804 ¢

error map: ⟨-0.070 +0.104 +0.188 -0.216]

• CWE: ~2 = 1200.0000 ¢, ~3375/3136 = 128.8871 ¢

error map: ⟨0.000 +0.173 +0.348 -0.085]

Optimal ET sequence: 121, 149, 270, 2309c, 2579c

Badness (Sintel): 2.35

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 117649/117612

Mapping: [⟨1 -4 2 -6 -9], ⟨0 52 3 82 116]]

Optimal tunings:

• WE: ~2 = 1199.9251 ¢, ~264/245 = 128.8818 ¢
• CWE: ~2 = 1200.0000 ¢, ~264/245 = 128.8890 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 1.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5], ⟨0 52 3 82 116 81]]

Optimal tunings:

• WE: ~2 = 1199.9510 ¢, ~14/13 = 128.8849 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.806

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5 -3], ⟨0 52 3 82 116 81 66]]

Optimal tunings:

• WE: ~2 = 1199.8754 ¢, ~14/13 = 128.8778 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8898 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.973

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125

Mapping: [⟨1 -4 2 -6 -9 -5 -3 5], ⟨0 52 3 82 116 81 66 -7]]

Optimal tunings:

• WE: ~2 = 1199.8964 ¢, ~14/13 = 128.8793 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.950

Witcher

See also: Amity family

Witcher tempers out the wizma (420175/419904) and the amity comma (1600000/1594323). It may be described as the 99 & 166 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 1600000/1594323

Mapping: [⟨1 -7 -20 -2], ⟨0 25 65 14]]

mapping generators: ~2, ~80/63

Optimal tunings:

• WE: ~2 = 1199.8698 ¢, ~80/63 = 412.0531 ¢

error map: ⟨-0.130 +0.283 -0.260 +0.178]

• CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0956 ¢

error map: ⟨0.000 +0.435 -0.100 +0.512]

Optimal ET sequence: 32c, 67c, 99, 265, 364, 463, 562, 1025

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 1240029/1239040

Mapping: [⟨1 -7 -20 -2 -35], ⟨0 25 65 14 112]]

Optimal tunings:

• WE: ~2 = 1199.9909 ¢, ~80/63 = 412.0758 ¢
• CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0788 ¢

Optimal ET sequence: 99e, 166, 265

Badness (Sintel): 2.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4375/4356, 10648/10647

Mapping: [⟨1 -7 -20 -2 -35 -43], ⟨0 25 65 14 112 136]]

Optimal tunings:

• WE: ~2 = 1200.0660 ¢, ~33/26 = 412.0928 ¢
• CWE: ~2 = 1200.0000 ¢, ~33/26 = 412.0709 ¢

Optimal ET sequence: 99ef, 166, 265

Badness (Sintel): 2.40

Semiwitcher

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 41503/41472, 585640/583443

Mapping: [⟨2 -14 -40 -4 -13], ⟨0 25 65 14 29]]

mapping generators: ~99/70, ~80/63

Optimal tunings:

• WE: ~99/70 = 599.9717 ¢, ~80/63 = 412.0766 ¢
• CWE: ~99/70 = 600.0000 ¢, ~80/63 = 412.0951 ¢

Optimal ET sequence: 32c, 166, 198, 364, 562, 926

Badness (Sintel): 2.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280

Mapping: [⟨2 11 25 10 16 24], ⟨0 25 65 14 29 53]]

Optimal tunings:

• WE: ~99/70 = 599.9666 ¢, ~33/26 = 412.0726 ¢
• CWE: ~99/70 = 600.0000 ¢, ~33/26 = 412.0944 ¢

Optimal ET sequence: 32cf, 166, 198, 364, 562, 926

Badness (Sintel): 1.57

Hitch

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 117649/117612, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31], ⟨0 50 130 28 -41]]

mapping generators: ~2, ~891/560

Optimal tunings:

• WE: ~2 = 1199.8455 ¢, ~891/560 = 805.9457 ¢
• CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0488 ¢

Optimal ET sequence: 67c, 198, 463, 661, 1124bde

Badness (Sintel): 3.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31 -134], ⟨0 50 130 28 -41 205]]

Optimal tunings:

• WE: ~2 = 1199.8415 ¢, ~891/560 = 805.9451 ¢
• CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0508 ¢

Optimal ET sequence: 198, 463, 661

Badness (Sintel): 2.51

Hitcher

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561

Mapping: [⟨1 -32 -85 -16 31 -1], ⟨0 50 130 28 -41 7]]

Optimal tunings:

• WE: ~2 = 1199.7690 ¢, ~78/49 = 805.8958 ¢
• CWE: ~2 = 1200.0000 ¢, ~78/49 = 806.0500 ¢

Optimal ET sequence: 67c, 198, 463f, 661ff

Badness (Sintel): 2.46